Demand side management: model-based control of aggregated power for

populations of thermostatically controlled loads

Cristian Perfumoa,c,∗, Ernesto Kofmanb, Julio H. Braslavskya, John K. Warda

aCSIRO Energy Technology, 10 Murray Dwyer Circuit, Mayfield West NSW 2304, AustraliabCIFASIS-CONICET, Department of Control, FCEIA-UNR Riobamba 245 bis (2000) Rosario, Argentina

cFaculty of Engineering and Built Environment, The University of Newcastle, NSW, Australia

Abstract

Large groups of electrical loads can be controlled as a single entity to reduce the power de-

mand in the electricity network. This approach, known as demand side management (DSM) or

demand response, offers an alternative to the traditional paradigm in the electricity market, where

matching supply and demand is achieved solely by regulating how much generation is dispatched.

Thermostatically controlled loads (TCLs) such as air conditioners (ACs), fridges, water and space

heaters are particularly suitable for DSM, as they store energy in the form of heat (or absence

of it), allowing to shift the load to periods where the infrastructure is less heavily used. DSM

can be implemented using feedback control techniques to regulate the aggregated power output

of a population of these devices. To achieve high performance, modern feedback controllers are

designed using a model of the plant (i.e. the population). However, most available models of a

population of TCLs are too complex to be used in control design. In this paper we present a

physically-based model for the aggregated power demand of a population of ACs and analytically

characterise its dynamic response to a step change in temperature set point. Such step response is

shown to display damped oscillations, a phenomenon widely known and simulated but, to the best

of our knowledge, not analytically characterised before. From this characterisation, a simplified

mathematical model is then derived and used to design an internal-model controller to regulate

aggregated power response by broadcasting set point offset changes. The proposed controller

achieves high performance provided the ACs are equipped with high resolution thermostats. With

∗Corresponding authorEmail address: cristian.perfumo@csiro.au (Cristian Perfumo)

Preprint submitted to Energy Conversion and Management September 7, 2011

coarser resolution thermostats, as typically used in commercial and residential ACs at present,

performance deteriorates significantly. We overcome this limitation by dividing the population

into clusters that receive a coarse-grained, potentially different, control signal. This clustering

technique recovers the performance achieved with high resolution thermostats at the expense of

some comfort penalty that can be quantified using the controller output.

Keywords: Demand side management, Load control, Air conditioning, Internal model control,

Load aggregation, Control signal quantization

1. Introduction

Demand side management (DSM) technologies include a wide range of strategies to reduce

power usage to balance supply and demand in electricity markets at peak periods, reducing the

pressure for upgrades in power generation and distribution infrastructure. Studies by the In-

ternational Energy Agency (IEA) suggest that DSM is more cost-effective and sustainable than

conventional policies based on supply side: each $1 invested in DSM has been estimated to off-

set $2 spent in supply side improvements, while contributing to reduce greenhouse emissions [1,

Chapters 7, 8]. As smart meters and appliances slowly become mainstream, DSM technologies

gain strength as an alternative for the electricity market: instead of increasing the generation to

satisfy customer needs, large groups of electrical devices may be controlled to reduce the demand

during critical periods [2, 3].

An important class of electric loads that can be integrated in DSM strategies with full respon-

siveness are thermostatically controlled loads (TCLs) [3, 4]. TCLs encompass devices such as air

conditioners (ACs), fridges, and space and water heaters, which are typically responsible for a

large proportion of the residential energy demand [5]. The flexibility of TCLs for demand control

comes as a result of their thermal inertia: TCLs may be viewed as a distributed energy storage

resource that can be controlled with constrains imposed by acceptable impact on end users.

The operation of TCLs can be manipulated by DSM for various reasons, the most common

one being to reduce the power demand during periods of high electricity prices or high electricity

demand. More recently, electricity markets are starting to consider the participation of loads

alongside conventional supply-side resources [3, 6], a trend that is likely to become more prominent

2

as aggregations of TCLs become more competitive with traditional supply side approaches [7, 8, 9].

Another application of DSM on TCLs is to track variations in generated energy, such as that

typically observed in renewable generation [4].

The aggregated power output of a population of TCLs may be effectively controlled by manip-

ulating a common temperature set point offset. As shown in [4, 5], small common offset changes in

temperature set point may be broadcast to the population to control their aggregate power with

minimal impact to individual users.

However, common changes in set point in a large population of TCLs may produce large

undesirable transients in their collective power response, as their states will tend to synchronise

under a common disturbance. Such transients are typically observed in TCLs when power supply

is interrupted for an extended period of time and then simultaneously restored, a phenomenon

traditionally known as cold-load pickup [10]. Similar undesirable collective behaviour may also

occur in large populations of loads under randomised (but uncoordinated) autonomous control, as

illustrated in [3] for populations of plug-in electric vehicles.

In the present paper we consider the design of a model-based, coordinated feedback strategy

to control aggregated power of ACs by manipulating common offset changes in their temperature

set points. Using modern control techniques, system dynamics may be controlled to high degrees

of performance by incorporating an accurate model of the uncontrolled dynamics in the design

of the controller, a notion referred to as internal model control [11]. A core component of the

control strategy proposed in this paper is the development of a model that accurately captures the

collective dynamics of TCLs and at the same time maintains a restricted mathematical complexity

that allows its systematic use in the design of the controller.

Substantial research on modelling populations of TLCs and using these models is available:

some authors concentrate on models based on first principles [12, 10, 13, 14, 15, 16, 17], others on

identifying the model parameters from a real population of devices [18, 19] and others on black-

box model identification techniques [4, 20]. However, the majority of these models are not simple

enough to be incorporated in a feedback controller.

For example, a well-known model developed by Malhame and Chong (M&C) [14, 15] consists

of a set of Fokker-Plank diffusion equations describing the probability density distribution of

3

temperature in a population of identical heaters that condition identical spaces. By integrating

the temperature distribution, the probability of a device being operating at a certain time (and

therefore the power usage profile of the population) can be calculated. Using M&C’s model in the

design of a feedback control algorithm is very difficult because this model is formulated as a system

of partial differential equations with no closed-form solution in the general case. This and other

difficulties such as communications cause that most of the approaches to control a population of

TLCs for DSM be in open loop [21, 17, 5].

One exception to open-loop approaches is [4], where Callaway describes the use of a broadcast

temperature set point offset as the input signal of a feedback-controller for the aggregated power

output of a population of ACs. Core dynamics of the model in [14] are captured in [4] by a

first-order linear ARMAX (AutoRegressive Moving Average eXogenous) model obtained using

standard black-box system identification techniques [22]. This identified model is then used in [4]

to develop a minimum variance controller that drives the power output of the population to track

the generated power of a wind farm.

The idea in [4] of using a feedback controller for the aggregated power demand of a population

of TCLs using a global temperature set point offset as the control signal has recently been adopted

in [23] and [24]. In [23], the authors incorporate a control signal to M&C’s model and develop

a Lyapunov-stable algorithm to control a population of homogeneous devices and rooms. In [24]

a homogeneous population is also considered to obtain an initial undamped model to which the

authors manually add a damping coefficient to adjust the modelled response to the simulated one.

The damped model is then used to develop a linear quadratic regulator controller.

Other type of control signal could be used to control the aggregated power output (such as

toggling the ON/OFF state of all the devices in certain temperature range [25]). However, the

benefit of a global temperature set point offset is that it directly relates to user comfort. This

relation easily allows the controlling entity (presumably the electricity utility) to estimate the end

user impact in a DSM scenario. One limitation of the control approach in [4, 23, 24] is that it relies

on temperature set point changes as little as 0.0025 oC, whereas the typical set point resolution of

hardware installed nowadays is two orders of magnitude higher, at around 0.1 to 0.5 oC.

The present paper formulates an aggregated model based on first principles that represents,

4

under certain assumptions, the power consumption of a population of ACs over time. We use this

model to show that the power response of a group of ACs to a common temperature set point

change presents underdamped oscillations, a phenomenon widely known and simulated but, to

the best of our knowledge, not analytically characterised before. From our analysis, we derive a

simplified reduced-order mathematical model that we use to design an internal-model controller to

regulate aggregated power response by broadcasting temperature set point offsets. The simulation

results show accurate load control performance, provided the temperature resolution of the ACs is

fine enough (similarly to [4]). As an implementation of the proposed control approach for coarse-

resolution ACs (such as the ones installed currently at most homes), we divide the population

into clusters that receive different coarse-resolution control signals following a method similar to

that of demultiplexing in communication networks. Thus, spatial diversification is exploited to

reduce the quantisation constraints on the aggregated control signal. We show that the proposed

clustering technique recovers the control performance of a common finely quantised control signal

at the expense of some comfort penalty. This penalty is quantified by estimating the predicted

percentage of dissatisfied people (PPD) [26] based on the temperature set point offsets used as the

control signal.

The remainder of the paper is organised as follows: Section 2 presents a well-known single-AC

model and describes how to use it to simulate a population of devices under a DSM scenario.

Section 3 presents our proposed model of a population of ACs. Section 4 develops the proposed

model-based controller for the population, and Section 5 refines the controller for implementation

on coarse-resolution thermostats. Section 6 shows how to quantify the comfort impact in the

proposed DSM scenario from the control signals sent to the clusters. Finally, Section 7 summarises

our conclusions.

2. A preliminary model for the power dynamics of an aggregation of ACs

We consider a population of ACs, each of which regulates the average temperature θ(t) of a room

by means of a thermostat and relay actuator with state m(t) ∈ {0, 1}, which determines whetherthe compressor is switched on (m = 1) or off (m = 0) according to a pre-specified hysteresis band

[θ−, θ+] and a desired temperature set point (θ− + θ+)/2. The dynamic behaviour of such an AC

5

can be more precisely described by the two-state hybrid mathematical model

dθ(t)

dt= − 1

CR[θ(t)− θa +m(t)RP + w(t)], (1)

m(t+) =

0 if θ(t) ≤ θ− + u(t)

1 if θ(t) ≥ θ+ + u(t)

m(t) otherwise,

(2)

where θa represents the ambient temperature (assumed constant), C and R are the thermal capac-

itance (kWh/oC) and thermal resistance (oC/kW) of the room being air-conditioned, and P (kW)

is the thermal power of the AC, which is the electrical power times its coefficient of performance.

The term w(t) in (1) represents unpredictable thermal disturbances (heat gains or losses) such as

the effect of people in the rooms, open doors and windows, and appliances. In (2), u(t) represents

a temperature set point offset that can be manipulated over time.

The hybrid model (1)–(2) has been used (with u(t) = 0) in early mathematical studies of

the dynamics of aggregated power of populations of ACs [12], [10], and more recently in [4], who

incorporated the temperature set point offset u(t) as an external control signal to achieve load

tracking to compensate the variability of wind energy generators. Note that controlling a group

of ACs using a set point offset avoids forcing a global absolute reference temperature for all the

devices, allowing the occupants to choose the temperature they are most comfortable with.

To describe the collective demand of a population of n ACs, let i ∈ [1, 2, . . . , n] denote the indexrepresenting the i-th device, which is described by a hybrid model of the form (1)–(2). Then, the

evolution of the aggregated normalised power demand D(t) (normalised by the maximum power

demand of the population) is given by the ratio

D(t) =

∑ni=1mi(t)Di∑n

i=1Di, (3)

where mi(t) represents the discrete state and Di the electrical power (Pi divided by the coefficient

of performance) of the i-th AC in the population.

The aggregation in (3) suggests a simple way to numerically simulate the evolution of the

aggregated power of such population of ACs by sampling values for the constant parameters C, R

and P for each AC from some pre-specified distributions and then computing and adding together

6

the individual power outputs [4]. Thus, the single-device model described by (1) and (2) has been

extensively used as a starting point for numerical studies of the behaviour of populations of TCLs

[12, 10, 14, 16, 17, 4].

In contrast, a mathematically precise analysis of (3) is challenging, since it comprises a distribu-

tion of n (potentially thousands) non-linear dynamic systems, each of which evolves independently

according to (1)–(2). One of the earliest and most important contributions to such analysis is the

work by Malhame and Chong [14], who developed a system of Fokker-Plank equations to describe

the time evolution of the probability distribution of an AC being at certain temperature. The

recent work by Callaway [4] refined the model developed in [14] and further simplified it for the

purposes of control design for load tracking of variable renewable generation.

The present paper uses the model in (1)–(2) both as a basis to develop a simplified analytical

model describing the aggregated power response of an entire population of ACs, and to perform

numerical simulations of large numbers of ACs. The latter are used to validate the proposed

simplified analytical model, presented in the following section.

3. Theoretical analysis towards a simple population model based on first principles

In this section we present an expression that describes how the proportion of ACs that are

switched on changes over time when a population of devices is excited with a step signal. More

importantly, we use this expression to derive a simplified model, suitable for use in control design.

We start by making the following simplifying assumptions about the continuous state of the

single-AC model in (1):

H.1 All of the ACs in the population have the same set point temperature θref = (θ+ + θ−)/2 and

the same hysteresis width θ+ − θ− = 1.

H.2 At t = 0, the temperatures are uniformly distributed in the interval [θ−, θ+].

H.3 The parameter C is distributed in the population according to some probability distribution.

H.4 For any AC, the temperature decreases when the AC is operating at the same rate it rises

when the device is switched off, which implies that RP = 2(θa − θref ) for each device.

7

H.5 θ+−θ− ≪ |θa−θref |. i.e., for each AC, the rate at which the temperature changes is constant,so that the temperature describes the triangular waveform as shown at the top of Figure 2.

H.6 The noise term ω(t) for each AC is negligible.

Assumptions H.1, H.3, H.4 and H.6 are a required simplification for our analysis. While

they may seem overly restrictive, in Section 5 we show that the controller designed using these

simplifying assumptions still can preserve good performance even when these assumptions are

relaxed.

H.4 implies that the duty cycle in steady state is 50%. In combination with H.1, H.4 also

implies that the parameter R is the same for all the ACs in the population, and the same applies

to P .

H.2 is reasonable since the temperatures of a group of devices that have been running indepen-

dently for long enough are distributed almost uniformly [4]. It is also sensible to make assumption

H.5 as the hysteresis width is expected to be significantly smaller than the difference between the

ambient and reference temperatures, rendering rates of temperature change constant rather than

variable (as obtained when (1) is solved [23]).

Finally, we assume H.6 because, when analysing a population of ACs as a whole, the variability

that ω(t) introduces is small compared to the one introduced by the fact that the parameter C is,

in general, different for each AC.

We analyse how a population of ACs, each governed by (1), (2) and H.1-H.6, reacts when a

step change of amplitude 0.5 is applied at t = 0, shifting the hysteresis boundaries to the right.

We will refer to these new boundaries as θpost− = θ− + 0.5 and θpost+ = θ+ + 0.5. Figure 1 shows the

temperature distributions just before and after the step. The top part of each subfigure represents

the ACs that are operating whereas the bottom part depicts the ones that are switched off. The

arrows indicate in which direction the temperatures are moving. We can see in Figure 1(a) and

1(b) that the ACs that were operating and had temperatures in [θ−, θpost− ] before the step switched

off after it.

Under assumptions H.1-H.6, the absolute value of the rate at which the temperature θi(t)

8

post +

ON

OFF

θ- θθ- θpost+

post +θ- θθ- θpost+

popula

tion d

ensit

y f

uncti

on

(a) Before step

ON

OFF

post +θ- θθ- θpost+

post +θ- θθ- θpost+

(b) After step

Figure 1: Distribution of temperatures before and after a 0.5oC step in the set point

changes for the i-th AC (as described in (1)) is given by the constant vi, defined by

∣

∣

∣

∣

dθi(t)

dt

∣

∣

∣

∣

≈ vi =θa − θrefCiR

. (4)

We introduce the variable

xi(t) = x0i + vit, (5)

where

x0i =

1 + θi(0)− θpost− if dθidt (0−) > 0,

θpost+ − θi(0) if dθidt (0−) < 0.(6)

Intuitively, we can say that xi(t) is the “unwrapped” mapping of the temperature θi(t). Figure 2

illustrates the equivalence between θi(t) and xi(t), the latter being a straight line, as implied by

H.4 and H.5. Note that in Figure 2, at the times the temperature θi(t) changes direction, xi(t)

reaches an integer value, changing from an even (on) to an odd (off) interval, or vice versa. We

refer to an interval of values of xi(t) as odd whenever xi(t) ∈ [2k+ 1, 2k+2], for k = 0, 1, . . . , andeven whenever xi(t) ∈ [2k, 2k + 1].

Now we are ready to formalise the step response of a population of ACs in the following

proposition (its proof can be found in Appendix A.1).

Proposition 1. For a population of ACs where the dynamics of each device are described by (1)

and (2), under the assumptions H.1-H.6, if the temperature set point of all the devices in the

9

postθ-

θpost+

x (t)

θ (t)

t

t

postθ-θ

post+ - =1

ON/OFF (per initial cond.)

ON

ON

OFF

OFF

0

1

2

3

4

5

i

i

Figure 2: Top: temperature θi(t) of one AC. Bottom: its “unwrapped” equivalent xi(t).

population is raised by 0.5 oC at time t = 0, the probability D(t) that a randomly-picked AC be

operating at time t is given by:

D(t) =Pr[x(t) < 1]

3+

∞∑

k=1

Pr[x(t) < 2k + 1]− Pr[x(t) < 2k] (7)

where Pr[·] is the probability operator.

Proposition 1 says that the probability that one randomly-picked AC be operating at time t is

given by the probability that x(t) is in an even interval plus a correction for the initial condition

arising from H.2. Note that for a large enough population, the probability (7) is equivalent to the

proportion of ACs in the population being operating at time t. Also, because all of the ACs have the

same power (H.4), this proportion is equal to the power consumption of all of the devices operating

at time t, normalised to the maximum demand (all of the ACs in the population operating), as

described in (3). Therefore, throughout this paper, we deliberately use the notation D(t) to refer

to the probability of a randomly-picked AC being operating, the proportion of operating ACs in

the population and the normalised power consumption.

In order to calculate actual values for (7), it is necessary to know how the temperatures change

for each AC in the population. This can be described by characterising the distribution of each

of the parameters R, P and C in the population. We adopt log-normal distributions for these

10

parameters, which have been used also in [4], as they are suitable for non-negative parameters and

have a moderate complexity of description.

Assuming that all of the ACs have the same thermal resistance R and the same thermal power

P, and the thermal capacitance C is distributed log-normally in the population, Eq. (7) can be

more explicitly calculated, as shown in the following corollary of Proposition 1.

Corollary 1. Under the assumptions of Proposition 1, let all of the ACs have the same thermal

resistance R and the same thermal power P, and let the thermal capacitance C be distributed log-

normally with mean µC and standard deviation σC . Then the speed v at which the temperature

changes (Equation (4)) is distributed log-normally and the ratio σrel between its standard deviation

σv and mean µv is

σrel =σvµv

=σCµC

. (8)

Furthermore, the probability D(t) that a randomly picked AC be operating at time t can then be

approximated by

D(t) ≈ 16+

1

6erf

[

log(1)− log(µx(0) + µvt))√2σrel

]

+1

2

∞∑

k=2

(−1)k+1erf[

log(k)− log(µx(0) + µvt))√2σrel

]

. (9)

where erf[·] is the Gauss error function and µx(t) is the mean of the values x at time t.

A formal proof of Corollary 1 is presented in Appendix A.2.

Figure 3 plots in dashed lines the expected power response according to (9), for different values

of σrel, to a 0.5oC step at t = 0. The output is normalised to the maximum power output (all

ACs turned on). The figure also shows, as a solid black line, the output to the same input when

we simulate using the PowerDEVs tool [27], 10000 ACs (according to (3)) assuming homogeneous

thermal resistance R and the same thermal power P for all the devices. The thermal capacitance

C is distributed log-normally in the population with a standard deviation to mean ratio of σrel.

The values for the parameters used for the simulations, detailed in Table 1, are the same as in [4],

except for the ambient temperature, which was adjusted to obtain a duty cycle of 0.5, to make a

comparison with the theoretical results possible. Figure 3 validates Proposition 1 and Corollary 1,

as the analytical and simulated responses are very close.

11

Note that in the simulated results, H.4, H.5 and H.6 were relaxed, considering a much more

realistic scenario. In particular, it is important to note that relaxing H.4 implies that, in general

terms, each room gets cooled down at a potentially different rate than it heats up (no fixed relation

between R and P). The aspect of H.4 that still holds for the simulated results is that the average

duty cycle in the population is 0.5, which is why the simulated results present a steady state

normalised power consumption of 0.5. The authors are currently working on an extension of the

analysis presented in this section to contemplate different duty cycles, which is a more realistic

scenario as, for example, a duty cycle higher than 0.5 would be expected for very hot days, and

a lower one for cooler days (especially for populations of oversized ACs, which are common).

Nevertheless, in Section 5 we show that the controller designed using the model obtained in the

present section, assuming a duty cycle of 0.5, shows robustness even when used on a plant with

an average duty cycle far from 0.5.

When all three parameters R, P and C are distributed, there is more variability in the speed

at which the temperature changes, and therefore the ACs tend to desynchronise faster (the sim-

ulated response with the parameters from Table 1, not shown, are more damped than those in

Figure 3). Nevertheless, the dominant dynamics are the same for both cases: the power response

of a population of ACs to a step in the temperature set point presents damped oscillations.

The expression (9) can be used to analytically show that the transients in aggregated power

due to a step change in temperature set point are characterised by underdamped oscillations. The

following proposition formalises this observation.

Proposition 2. Under the assumptions of Corollary 1, the initial transients in the response of the

aggregated power to a step change in temperature set point displays oscillations with period

T ≈ 2/µv, (10)

and peaks with amplitudes A(t) that decay with time according to the approximate envelope bound

1− erf(1/z(t)) ≤ A(t) ≤ erf(1/z(t)), (11)

where

z(t) = 2√2σrel(µx(0) + µvt− 1/2). (12)

12

−500 0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

analytical

simulated

−500 0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

analytical

simulated

−500 0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0analytical

simulated

−500 0 500 1000 1500 20000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0analytical

simulated

σrel = 0.05 σrel = 0.1

σrel = 0.2 σrel = 0.5

t (minutes)t (minutes)

t (minutes)t (minutes)

DD

DD

Figure 3: For different values of σrel, expected and simulated D (normalised power demand) response to an increase

of 0.5 oC in all ACs’ temperature set points. The simulated results are for 10000 ACs with log-normally distributed

C and constant R and P , and were obtained using the parameters from Table 1. To obtain a duty cycle of 0.5, θa

was set to θref +RP/2 = 34oC.

13

Table 1: Simulation parameters.

Parameter Value Description

R 2 oC/kW Mean thermal resistance

C 10 kWh/oC Mean thermal capacitance

P 14 kW Mean thermal power

θ− 19.5oC Lower end of hysteresis band

θ+ 20.5oC Higher end of hysteresis band

θa 32oC Ambient temperature

σw 0.01 Standard deviation of the noise process w in Eq. (1)

σrel 0.05/0.1/0.2/0.5 Standard deviation of log-normal distributions as a fraction of

the mean value for R, C and P

A formal proof of Proposition 2 can be found in Appendix A.3.

Figure 4 depicts the envelope bound (11) of the oscillations in the step power response along

with the expected power output calculated from (9) for different values of σrel as a function of the

offset and rescaled time z, as defined in (12), which allows us to plot all responses within a common

envelope. Each curve starts at the corresponding value of z for t = 0. We can see in Figure 4 that

for a range of values of σrel, the bounds in (11) capture accurately the envelope of the response.

We also observe in Figure 4 that for values of z larger than 1.8, the bounds in Equation (11) are no

longer valid (see proof of Proposition 2). Nevertheless, the fundamental dynamics of the response

(i.e., damped oscillations) are still observed for larger values of z, and hence t.

The analysis carried out so far demonstrates that the step response of the aggregated power of

a population of ACs under the assumptions H.1-H.6 is dominated by decaying oscillations, which

corroborates the simulation results reported by a number of authors [10, 15, 13, 4]. In particular, if

the duty cycle is around 0.5, the “mountains” and “valleys” of D(t) have equal semi-periods, which

suggests that second order dynamics would be a suitable approximation of the aggregated power

response. The following corollary to Proposition 2 provides formulas to compute the characteristic

14

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

z (offset and rescaled time)

D

σrel = 0.05σrel = 0.1σrel = 0.2

Figure 4: Envelope of the power peaks. The dotted and dashed lines represent the normalised power demand D

for different values of σrel according to (9), all of them adjusted to the offset and rescaled time variable z. The

continuous line is the envelope given by (11)-(12).

polynomial of such second order dynamics directly from the parameters describing the population

of ACs (i.e. Table 1).

Corollary 2. Under the assumptions of Proposition 2, the response of a population of ACs to a

0.5 oC step can be modelled as

D(t) = Dss(θref ) + L−1{Gp(s)0.5/s} (13)

where Gp(s) is a second order linear, time-invariant (LTI) system characterised by the transfer

function

Gp(s) =b2s

2 + b1s+ b0s2 + 2ξωns+ ω2n

(14)

whose parameters are

ξ =log(r)

√

π2 + log2(r); ωn =

πµv√

1− ξ2;

b0 =ω2n(Dss(θref + 0.5)−Dss(θref))

0.5; b1 = 0.5µv + 2b2ξωn and b2 = Dss(θref). (15)

where

r =|erf( 1

0.9+2√2σrel

)− 0.5||erf( 1

0.9)− 0.5|

15

and

Dss(θ) =

(

1 +log(1 + H

θa−θ−H/2)

log(1 + HPR+θ−θa−H/2)

)−1(16)

Corollary 2 (whose proof can be found in Appendix A.4) provides direct, rule-of-thumb, in-

formation about the step response of a population of ACs. For example, using Corollary 2 we

compute ξ = 0.259 and ωn = 0.033 for the scenario shown in the bottom-left plot in Figure 3.

These values of ξ and ωn give an estimated 1% settling time ts = 4.6/(ξωn) = 525 (minutes)

[28], which means that, for the considered scenario, the step response will approximately decay

to 1% in 525 minutes. By comparison with the simulated response of 10000 ACs shown at the

bottom-left plot in Figure 3, we can see that the settling time computed using Corollary 2 is an

excellent estimate of the actual settling time of the collective response. Note that to compute such

an estimate otherwise, we would need to perform one of these two computationally intensive tasks:

(a) run numerical simulations of a large number of ACs (such as the ones shown in solid black in

Figure 3) or (b) approximate numerically M&C’s set of Fokker-Plank differential equations [14].

More importantly, knowing the transfer function (14) allows us to design a model-based con-

troller for the a population of ACs based only on the physical parameters of such a population.

Such a controller is presented in the following section.

4. Controlling the plant

In this section we present the design of a controller for a population of ACs that is able to

adjust the power output to a desired profile by manipulating the temperature set points of the

devices. We use a model-based control structure known as internal model control (IMC) [11].

Such a model-based control structure integrates a model of the plant as a fundamental part of

the controller. The following three subsections describe the model of the plant for control design

purposes, its implementation in the design of the controller, and present the closed-loop responses

obtained under different simulated scenarios.

16

4.1. A linear, time-invariant model for a population of ACs for control design

We use the second order linear, time-invariant (LTI) model structure in (14) as a model for the

aggregated response.

Table 2 shows the values of (15) obtained for a population of ACs described by the parameters

in Table 1 (σrel = 0.2). This combination of parameters of the population is the same as in [4],

except for hysteresis width, for which we use 1 oC as opposed to 0.5 oC, as we consider 0.5 oC

overly tight.

We also show in Table 2 the parameters obtained when a second order model is manually

identified from the simulated response to a 0.5 oC step of a population of 10000 ACs as described

in (3) with parameters as per Table 1 (σrel = 0.2). We fit the parameters of the proposed second

order model structure by measuring peaks, period and steady state gain in the simulated step

response of 10000 ACs.

Table 2: Transfer function parameters of the second order model in (14) calculated from (15) and identified manually.

Method ξ ωn b0 b1 b2

As per (15) 0.259 0.033 3.70× 10−5 0.029 0.446Manually identified 0.3505 0.0326 3.61× 10−5 0.038 0.45

Figure 5 compares the simulated response with that of the calculated second order LTI model

using (15), and the one manually identified. The figure shows that the calculated model can

capture the dominant dynamics of the simulated response very closely for the considered set of

parameters. Moreover, from Table 2 and Figure 5 we can conclude that the second order model

calculated from the parameters of the population of ACs is comparable to a manually identified

model. Thus, throughout the reminder of this paper we will use the calculated model using (15)

for our control design.

Note that, as per Table 1, we simulate a population where the parameters P, R and C are

all distributed lognormally (σrel = 0.2 indicates that the standard deviation of each parameter

is calculated as its mean times 0.2), which relaxes assumptions H.3 and H.4. In fact, all the

simulated results that we present in the reminder of this paper are obtained from populations

17

where P, R and C are all distributed. We do this to illustrate the robustness of the proposed

control algorithm, which preserves the desired performance even when we move away from the

simplifying assumptions on which the controller is based.

0 500 1000 15000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

simulated

manually identified

analitically calculated

t (minutes)

D

Figure 5: Step power response D for 10000 simulated ACs with parameters described in Table 1 (σrel = 0.2) and

using the second order systems calculated with (15) and identified manually.

4.2. Using the calculated LTI model to control the population via internal model control

Figure 6 shows a typical IMC structure. The input signal Dref(t) to the controller represents

the desired normalised aggregated power (proportion) of operating ACs in the population at time

t. Given this reference signal and the actual normalised aggregated power D(t) observed in the

population, the controller computes the temperature set point offset signal u(t) to be broadcast to

the ACs.

G (s)d G (s)p-1

G (s)p

PlantD(t)D (t)ref

-

-

IMC Controller

u(t)

Figure 6: Internal model control structure.

18

The IMC controller encompasses a model of the plant Gp(s), its inverse (or a stable approx-

imation if the inverse does not exist) G−1p (s), and the target desired behaviour in closed loop

represented by its transfer function Gd(s). For simplicity, we propose

Gd(s) =1

Tcs+ 1

where Tc is the closed-loop time constant of the system. For the results presented in this paper we

chose Tc = 2 (minutes), since it allows reaching the steady state reference value within a reasonable

time (10 minutes) using an affordable amount of control effort.

4.3. Preliminary closed-loop performance and implementation issues

We propose a DSM scenario with two stages: power reduction and comfort recovery. During

power reduction, the controller manipulates the temperature set points to achieve an aggregated

power output lower than the starting steady state value. In recovery, the demand raised over the

steady state value to make up for the increase in the temperatures of the rooms caused during the

reduction period.

0 200 400 600 800 1000 12000.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

Nor

mal

ised

dem

and

D(t)

Dref(t)

D(t)

Dref(t)

D(t)

Dref(t)

0 200 400 600 800 1000 1200−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

t (minutes)

u(t)(C

0)

(a) ∆u = 0.05

0 200 400 600 800 1000 12000.00.10.20.30.40.50.60.70.80.91.0

Nor

mal

ised

dem

and

D(t)

Dref(t)

0 200 400 600 800 1000 1200−1.0−0.8−0.6−0.4−0.20.00.20.40.60.81.0

t (minutes)

u(t)(C

0)

(b) ∆u = 0.5

Figure 7: DSM scenario on a population of 10000 ACs between t = 100 and t = 580 (minutes). The control aims

to make the normalised power output D(t) follow the reference signal Dref(t) (top plot). The control control signal

u(t) is shown in the bottom plots. The quantisation of u(t) is ∆u = 0.05 for (a) and ∆u = 0.5 for (b).

An instance of this scenario is presented in Figure 7(a). The ACs run uncoordinated (global

control u(t) = 0) for the first 100 minutes, with a steady state power output of 44%, relative to

19

the maximum. The reduction period takes place between t = 100 and t = 340 (minutes), reducing

the power output by 10% of the maximum load. The recovery takes place between t = 340 and

t = 580 (minutes), when the power output is maintained at 54% (the steady state output plus

10% of the maximum). The DSM scenario finalises at t = 580, when the ACs continue to operate

without global control.

This scenario is far from optimal for DSM. The oscillations in D observed after t = 580

indicate that, once the devices stop being controlled, the system is not left in steady state. One

could control the system after t = 580 in order to gradually bring D back to steady state, but it

should be kept in mind that the control period should be finite, as it is unlikely for a customer to

accept that their ACs be manipulated indefinitely. It is not the objective of the present paper to

explore optimal recovery phases but rather to show a way to control a population of ACs to follow

a desired reference signal.

Figure 7(b) presents the same scenario as Figure 7(a) except for a coarser granularity of the

control signal. Because the temperature sensors in the ACs do not have infinite resolution and

accuracy, the input signal must be quantised for the control to be implementable. We observe in

Figure 7(b) that for a coarse quantisation ∆u = 0.5, the controller is unable to track the reference

since, as we saw in Figure 3, a step of 0.5 oC implies a large (transient) change in the output.

Conversely, when we use a granularity small enough ∆u = 0.05, the controller successfully makes

the plant follow the reference signal (Figure 7(a)). Thus, the granularity of the signal plays a key

role in the quality of the output.

From the results presented in Figure 7 arises the following implementation difficulty: how can

DSM be implemented using a temperature set point change as the control signal in a feedback-

controlled system, if small values of ∆u are needed for an acceptable response (as also suggested

in [4]) and, on the other hand, the set point resolution of the majority of residential ACs is 0.1,

0.5 or even 1 oC? In the following section we propose a practical solution to this issue.

20

5. Cluster-based control implementation: exploiting spacial diversification to solve

the problem of controlling currently installed ACs

To resolve the temperature set point resolution issue pointed out in the previous section, we

propose to divide the population of ACs into clusters, and implement a mapper that demultiplexes

the fine-granularity control signal for the entire population into multiple coarse-granularity control

signals, one for each cluster defined. Figure 8 presents a schematic diagram of how the mapper

interacts with the clusters and the controller. Note that no changes in the controller are needed,

since the mapper is considered part of the plant. The mapper receives the “global” control signal

u(t), which is discretised very finely to a small quantum ∆u (e.g. 0.05 oC) and generates L different

“cluster” signals ui(t), which are multiples of a coarse quantisation ∆′u that the ACs can deal with

(e.g. 0.5 oC). If we choose L∆u to be a multiple of ∆′u, each cluster signal is defined1 by:

ui(t) =

floor[u(t)∆′u

]∆′u if i > L mod (u(t),∆′u)

∆′u

floor[u(t)∆′u

]∆′u+∆′u if i ≤ L mod (u(t),∆′u)∆′u

.

(17)

Mapperu(t)

u (t)1

u (t)2

u (t)L

Cluster 1

Cluster 2

Cluster L

...

D (t)1

Σ

D (t)2

D (t)L

D(t)IMC

Controller

D (t)ref

Plant

Figure 8: Redefinition of the plant when incorporating a mapper to translate the global control signal to several

different signals, one per cluster.

For example, if the population is divided in 10 clusters and u(tm) = 1.15 at t = tm, then the

cluster control signals will be ui(tm) = 1.5 for i = 1, 2, 3 and uj(tm) = 1.0 for j = 4, ..., 10.

1Fairness problems such as the one in (17), where the clusters with lower numbers always receive a larger

temperature set point offset, are easily overcome by assigning virtual numbers to the clusters and changing these

numbers every time a DSM scenario takes place.

21

Figure 9 shows the same DSM scenario as Figure 7: 10000 ACs being controlled to reduce the

load from 44% to 34% between t = 100 and t = 340 (minutes) and to increase it to 54% between

t = 340 and t = 580. In Figure 9 the population is divided in 10 clusters of 1000 ACs each, using

(17) to generate the 10 signals. We use a fine-grain ∆u = 0.05oC and coarse-grain ∆′u = 0.5oC.

The bottom plot in Figure 9 shows the global control signal u(t) as well as the individual signals for

the first and tenth clusters u1(t) and u10(t). From (17), it follows that the rest of the cluster control

signals (u2(t), u3(t),...,u9(t)) are bounded above by u1(t) and below by u10(t). Also, note that at

any time t, |u(t) − ui(t)| < ∆′u and |ui(t) − uj(t)| ∈ {0,∆′u}, which is to say that the distancebetween any cluster control signal and the global control signal is less than the quantisation ∆′u,

and the distance between two individual signals is either 0 or ∆′u.

0 200 400 600 800 1000 12000.0

0.1

0.2

0.3

0.4

0.5

0.6

Nor

mal

ised

dem

and

D(t)

Dref(t)

0 200 400 600 800 1000 1200−0.5

0.0

0.5

1.0

u(t)

u1(t)

u10(t)

t (minutes)

u(t)(C

0)

Figure 9: Normalised aggregated power demand D (top) and global and cluster-level input control signals u(t),

u1(t), u10(t) (bottom) for a DSM scenario between t = 100 and t = 580 (minutes) on a population of 10000 ACs.

The control aims to make the normalised power output D(t) follow the reference signal Dref(t). The 10000 ACs are

divided into 10 clusters of 1000 ACs each. The bottom plot shows the global control signal (solid) as well as the

individual signals for clusters 1 and 10 (dotted and dashed respectively).

Clustering does not hurt output performance, as the variance of the difference between the

reference and the output in Figure 9 is slightly smaller than that of Figure 7(a) (no clusters),

for both the reduction and recovery phases. However, there is a comfort penalty associated with

clustering, as in the example in Figure 9 each AC receives a set point change up to ten times larger

22

than that of the scenario in Figure 7(a). We quantify this impact on comfort in Section 6.

In Figures 7(a) and 9, we can see that our controller performs well even when the parameters

P, R and C are distributed in the population, relaxing assumption H.4. Figure 10 shows a scenario

that includes this relaxation and also considers a duty cycle far from the 50% implied by H.4. We

simulate such a scenario with the same parameters as that from Figure 9, except for the ambient

temperature, which is set to 45 oC, causing a duty cycle in steady state of 85%. The control in

Figure 10 aims to lower the demand to 75% in the reduction period and increase it to 95% during

the recovery period. Note that the controller was designed with the model we calculated using (15)

with the population parameters in Table 1, which assume a duty cycle close to 50%. Yet, when

we use the resulting controller for a plant with very high duty cycle, the output remains close to

the reference, showing the robustness of the proposed control design.

0 200 400 600 800 1000 12000.500.550.600.650.700.750.800.850.900.951.00

Nor

mal

ised

dem

and

D(t)

Dref(t)

0 200 400 600 800 1000 1200−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

t (minutes)

u(t)(

oC)

Figure 10: Normalised output power demand D (top) and global input control signal u(t) (bottom) for a DSM

scenario between t = 100 and t = 580 (minutes) on a population of 10000 ACs with 85% mean duty cycle. The

control aims to make the normalised power output D(t) follow the reference signal Dref(t). The 10000 ACs are

divided into 10 clusters of 1000 ACs each. The bottom plot shows the global control signal.

Figure 11 shows the power output of clusters 1 and 10 during the scenario presented in Figure 9.

Each curve is normalised to the maximum power of the cluster it represents. The demand peaks

at the cluster level in Figure 11 evidence that if the clusters had to be defined geographically (e.g.

a group of adjacent city blocks), the presented algorithm may not be suitable, since high peaks

23

0 200 400 600 800 1000 12000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cluster 1

Cluster 10

t (minutes)

D

Figure 11: Output power demand D for two clusters (each normalised to maximum demand of cluster) under the

DSM scenario depicted in Figure 9.

might appear in the infrastructure powering that geographical area. Alternatively, the clusters can

be defined so that each geographical area includes devices from many or all of the clusters. If such

a cluster layout is possible (e.g. communications-wise), a peak at the cluster level will not have

any negative impact on the distribution infrastructure.

Another consideration when defining the clusters is the distribution of the AC parameters in

each cluster. Our analysis assumes that the ACs are randomly chosen from the population to form

the clusters and therefore each cluster shares the statistical properties of the population (e.g. same

mean C). If a different clusterisation (e.g. some clusters with high and some other with low mean

thermal capacitance) is required, additional analysis should be carried out in order to determine

how this heterogeneity affects the results presented in this section.

6. A method to quantify the trade-off between discomfort and controllability

One advantage of using temperature set point offset as a control signal is that the entity imple-

menting the DSM (e.g. the utility) is able to obtain an estimate of the comfort impact straight-

forwardly (just by observing ui(t)), since the temperature of any AC in the cluster i converges

towards the hysteresis band [θ− + ui(t), θ+ + ui(t)].

24

It is apparent, then, that changes in ui(t) will have an impact on the comfort of the occupants in

the i-th cluster. One of the ways of quantifying that impact is by observing the Predicted Percent

Dissatisfied (PPD), a discomfort metric defined by the American Society of Heating, Refrigerating

and Air-Conditioning Engineers (ASHRAE) [26] that estimates the percentage of people that would

vote that they are uncomfortably cold or hot if they were surveyed. The PPD is calculated using

the Predicted Median Vote (PMV), which in turn is a function of temperature, clothing, humidity,

activity level of the occupants, air velocity and other parameters [26].

In this section we propose a generalisation of the PPD that allows us to estimate the comfort

range of the occupants of all the conditioned spaces in one cluster, regardless of the individual set

point temperatures of each AC.

We assume that the occupants have chosen the set point they are most comfortable with. In

other words, ui(t) = 0 achieves the minimum dissatisfaction (PPD) possible2. Figure 12 plots

the range spanned by a set of PPD curves representing different combinations of clothing level,

occupant’s activity, air velocity and relative humidity according to the values shown in Table 3.

All of the curves in Figure 12 were superimposed, making their minima coincide at u(t) = 0, and

the range that they collectively span was shaded, representing the range of comfort impact for a

value of ui(t). In other words, by looking at the maximum and minimum values of ui(t) during a

DSM scenario, we can get an upper bound of how uncomfortable occupants get during that period.

Table 3: Parameter range for calculating PPD

Parameter Value Description

Clothing level {0.36, 0.61} (clo) 0.36 = Walking shorts, short-sleeveshirt. 0.61 = Trousers, long-sleeve

shirt

Metabolic heat generation {1, 2} (met) 1 = reading/writing. 2 = mild house-cleaning

Air velocity {0, 0.2} (m/s)

Relative humidity {40, 60, 80} (%)

2Note that the theoretical minimum PPD is 5%, modelling that it’s impossible to keep everyone happy.

25

X: 1Y: 10.15

u(t) (C)

X: 0.5Y: 6.37

PP

D (

%)

−2 −1 0 1 20

5

10

15

20

25

Figure 12: Range of change in predicted percent dissatisfied when the temperature set point is changed for the

values for clothing level, metabolic heat generation, air velocity and relative humidity presented in Table 3.

For example, from Figure 9, we see that max(u1(t)) = 1 and max(u10(t)) = 0.5. By looking

at Figure 12 one can conclude that in the worst case (assuming that the maximum value of ui(t)

were maintained for long enough so that the temperature catches up with it), the PPD of the set

point temperature would have risen from 5% to 6.37% for cluster 10 and to 10.15% for cluster 1.

Because u1 and u10 are boundaries for any other individual signal, their comfort impacts bound

that of any other cluster. An alternative way of calculating these boundaries without looking at

the cluster signal is by “ceiling” the global control signal u(t) to the closest multiple of ∆′u for the

upper bound (i.e. u1(t)) and “flooring” it for the lower bound (i.e. u10(t)).

To the best of our knowledge, this is the first attempt to quantify the comfort impact when a

population of ACs are controlled for DSM by temperature set point offset changes. Applying the

method presented in this section, a controller can be designed to regulate the power demand to

satisfy a given constraint on maximum comfort impact.

7. Conclusions

In this paper we have considered modelling and control of the aggregated power demand of a

population of ACs. The ACs independently regulate temperature using relay-type actuators and

a thermostat with a hysteresis band around a temperature set point.

Our model characterises the aggregated power consumption of the population of ACs by de-

scribing how the proportion of operating ACs varies over time, following a step change in the

26

temperature set point of all the ACs. The ACs are assumed to be operating in steady state before

the step change.

Under the additional assumption that the thermal capacitances are distributed log-normally in

the population, we derived an explicit formula for the transient response to the common set point

offset, which we further characterise as an underdamped oscillatory response. To the best of our

knowledge, this is the first work that presents mathematical approximations for the period and

amplitude envelope of such response. These approximations are shown to satisfactorily capture

the dynamics of the response of a numerically simulated population of 10000 ACs over a realistic

range of parameter values. Moreover, we presented a way to derive the transfer function of an LTI

second order system that characterises the aggregated power response of the population. Not only

does this model provide rule-of-thumb information about the response (e.g. settling time) with no

need for intensive numerical computations, but it is also apt for control design.

We used such a model to design an internal model control structure to compute common

temperature set point offsets to be broadcast to the ACs as a control signal. We demonstrated that

using this approach, the aggregated power output of the population of ACs can be satisfactorily

controlled to provide a reduction in the demand over a pre-specified period of time. The proposed

model-based control displays robust performance even when the parameters characterising the

population violate the simplifying modelling assumptions.

The proposed controller relies on fine resolution changes to the temperature set point of the

ACs. In cases where the ACs only accept coarse set point offsets (for example 0.5 oC) we presented

an implementation technique based on dividing the population into logical clusters and sending

each cluster different control signals, each of them with admissible resolution. The aggregated

power output does not show performance degradation compared to using a global, fine-grained,

signal. Clustering, however, does incur in a comfort penalty that in a practical design needs to be

deliberately traded off and can result in localised demand peaks if not appropriately distributed

across the electricity network. Our method to quantify the range of discomfort for a given input

signal enables the design of more advanced controllers that explicitly take into account the comfort

impact associated with the control signal.

One of the lines we would like to explore in the future is the feasibility of using Corollary 2 to

27

determine the parameters of a population from a system identification experiment (i.e., from the

step response of a population of ACs, identify a second order system and from its transfer function

determine the parameters of the population).

The proposed approach provides a pathway to cost-effectively manage network demand, poten-

tially deferring upgrades in the electricity transmission and distribution infrastructure. Moreover,

this research is directly applicable to other types of TCLs such as fridges, cool rooms, water and

space heaters.

Appendix A. Proofs and derivations

Appendix A.1. Proof of Proposition 1

Proof. From (5) and (6), the operational state mi(t) of the i-th AC in the population is defined by

mi(t) =

0 if 2k − 1 ≤ xi(t) < 2k

1 if 2k ≤ xi(t) < 2k + 1,(A.1)

for k = 1, 2, . . .

In other words, we can determine whether or not the i-th AC is turned on based on xi(t) being in

an odd or even interval, if xi(t) ≥ 1. This is, however, not the case if xi(t) < 1. When (6) is appliedto every point in the temperature distributions after the step change shown in Figure 1(b), the

distribution of the initial values of x(t) for all of the ACs, namely x(0), is as shown in Figure A.13.

Note that, initially, only one third of the ACs that satisfy xi(0) < 1 are turned on. Therefore,

under assumptions H.1-H.6, the probability that a randomly-picked AC is operating at time t given

x(t) < 1 is

Pr[m(t) = 1|x(t) < 1] = Pr[x(t) < 1]3

. (A.2)

Thus, for a number of ACs sufficiently large, the proportion D(t) of operating ACs in the

population at time t is equivalent to the probability of an AC being operating at time t; namely,

D(t) =Pr[x(t) < 1]

3+

∞∑

k=1

Pr[x(t) < 2k + 1]− Pr[x(t) < 2k].

28

0 1.0 1.5 2.0 2.5 3.0x(0 )

ON/OFF interval

(per initial cond.)

ON

interval

OFF

interval

0.5

ONOFF

+

Figure A.13: Distribution of x(0+) (immediately after the temperature set point step change at t = 0).

Appendix A.2. Proof of Corollary 1

Proof. The fact that v is log-normally distributed follows from (4) and basic properties of the

expected value and standard deviation of a log-normal distribution.

Let us now show the equality σv/µv = σC/µC in (8). Considering that in (4) only the parameter

C is distributed in the population (because of H.1, H.3, H.4), we have that

σvµv

=s.d.[

θa−θrefCR

]

E[θa−θref

CR]

=s.d.[1/C]

θa−θrefR

E[1/C]θa−θref

R

=s.d.[1/C]

E[1/C](A.3)

where s.d.[·] and E[·] are the standard deviation and expected value operators. Thus, since C islog-normally distributed,

s.d.[1/C]

E[1/C]=

s.d.[C]

E[C]=

σCµC

= σrel. (A.4)

From the fact that v has a log-normal distribution, it follows from (5) that x(t) will be approx-

imately log-normal for t large enough. Therefore, assuming that x(t) is distributed log-normally,

we have

Pr[x(t) ≤ y] = 12+

1

2erf

[

log(y)− µ(t)√

2σ2(t)

]

(A.5)

where

µ(t) = log(µx(t))−1

2log

(

1 +σ2x(t)

µ2x(t)

)

and

σ2(t) = log

(

1 +σ2x(t)

µ2x(t)

)

.

29

That is, µ(t) and σ(t) are the mean and standard deviation of the normal distribution associated

to the log-normal distribution characterising x.

According to (5), the mean µx(t) and variance σ2x(t) of x(t) evolve as follows:

µx(t) = E[x(t)]

= E[x(0)] + E[v]t = µx(0) + µvt, (A.6)

σ2x(t) = (s.d.[x(t)])2

= (s.d.[x(0)])2 + (s.d.[v])2t2 = σ2x(0) + σ2vt

2, (A.7)

Thus, we can approximate, for t large enough,

σ2x(t)

µ2x(t)≈ σ

2v

µ2v= σ2rel. (A.8)

and then, using (A.6) and (A.8) in (A.5) we obtain

Pr[x(t) ≤ y] ≈ 12+

1

2erf

[

log(y)− log(µx(0) + µvt) + 12 log(1 + σ2rel)√

2 log(1 + σ2rel)

]

. (A.9)

We can approximate log(1 + σ2rel) ≈ σ2rel with an absolute error of the order of (σrel)8/2, arrivingto

Pr[x(t) ≤ y] ≈ 12+

1

2erf

[

log(y)− log(µx(0) + µvt)√2σrel

]

(A.10)

were we have disregarded the term 12log(1 + σ2rel) from the numerator inside the erf function in

(A.9) because it is negligible as compared to log(µx(0) + µvt)).

Replacing (7) with (A.10) for y = 1, y = 2k and y = 2k + 1, we obtain (9).

Appendix A.3. Proof of Proposition 2

Proof. The mean µx(t) and variance σ2x(t) increase monotonically with time and, in fact, after a

short initial period of time, only one value of k will account for the dominant contribution to the

summation term in (9). Thus, we approximate the probability of randomly-chosen AC satisfying

30

k < x(t) < k + 1 as

Pr[k < x(t) < k + 1] = Pr[x(t) < k + 1]− Pr[x(t) < k]

≈erf(

log(k+1)−τ√2σrel

)

− erf(

log(k)−τ√2σrel

)

2, (A.11)

where

τ = log(µx(0) + µvt). (A.12)

For even values of k, there will be a positive peak of power when t is such that (A.11) is a local

maximum. The peak will have amplitude Dk/∑n

i=1Di ≈ Pr[k < x(t) < k+1]. Analogously, whenk is odd, a negative peak of power will be encountered when t is such that (A.11) is maximum and

its amplitude will be Dk/∑n

i=1Di ≈ 1− Pr[k < x(t) < k + 1].Therefore, for both even and odd values of k we need to find the value of τ for which (A.11)

is maximized in order to compute the amplitude of the peaks. For that value of τ , it should be

verified that

d

dτ

erf(

log(k+1)−τ√2σrel

)

− erf(

log(k)−τ√2σrel

)

2

= 0

from which we obtain(

log(k + 1)− τ√2σrel

)2

=

(

log(k)− τ√2σrel

)2

and thereforelog(k + 1)− τ√

2σrel= ± log(k)− τ√

2σrel. (A.13)

The equality (A.13) can only be satisfied when its right hand side is negative, which yields the

value of τ for the k-th peak in the output:

τ ∗(k) =log(k + 1) + log(k)

2. (A.14)

Replacing (A.14) in (A.11) and operating we obtain

maxt

Pr[k < x(t) < k + 1] ≈ erf(

log(k + 1)− log(k)2√2σrel

)

= erf

(

log(1 + 1/k)

2√2σrel

)

.

For values of k > 1, we can approximate log(1+1/k) ≈ 1/k with an absolute error of the orderof 1/(2k2), and then

maxt

Pr[k < x(t) < k + 1] ≈ erf(

1

2k√2σrel

)

. (A.15)

31

On the other hand, from (A.12) and (A.14) we know that

τ ∗(k) = log(µx(0) + µvt∗(k)) =

log(k + 1) + log(k)

2= log(

√k2 + k),

from which we obtain

µx(0) + µvt∗(k) =

√k2 + k, (A.16)

where t∗(k) is the time of the k–th peak.

For k > 1 we can approximate√k2 + k ≈ k + 1

2with an absolute error of the order of 1/(8k)

using the first two terms of the Taylor series expansion. Thus, from (A.15) and (A.16) we obtain

maxt

Pr[k < x(t) < k + 1] ≈ erf(

1

2k√2σrel

)

≈ erf(

1

2(µx(0) + µvt∗(k)− 12)√2σrel

)

.

(A.17)

This last equation says that the k–th peak of power occurs when µx(0) + µvt− 1/2 = k, whichis to say that t∗(k) = (k + 1/2 − µx(0))/µv. Therefore, the period of the oscillations is given byt∗(k + 2)− t∗(k) = 2/µv, which shows (10).

Finally, inequality (11) is shown by evaluating the amplitude of the k–th peak, which is

E[Dk/

n∑

i=1

Di] ≈ erf(

1

2k√2σrel

)

, if k is even (A.18)

or

E[Dk/

n∑

i=1

Di] ≈ 1− erf(

1

2k√2σrel

)

, if k is odd. (A.19)

Taking into account that erf(1/z) decreases monotonically for positive values of z, expressions

(A.18) and (A.19) show how the amplitude of successive peaks decreases with k. Moreover, the

larger σrel, the faster it decreases.

Appendix A.4. Proof of Corollary 2

For a given value of z = z1, the upper part of the envelope of the oscillatory damped step

response is erf(1/z1) according to (11). After one semi-period 1/µv (see (10)), we have z2 =

32

z1(t + 1/µv) and, applying (12), z2 = z1 + 2√2σrel. Thus, the ratio between the distance of two

successive peaks to the steady state value (0.5), namely Dk+1 and Dk is:

Dk+1/Dk =|erf( 1

z1+2√2σrel

)− 0.5||erf( 1

z1)− 0.5| . (A.20)

As seen in Figure 4, the approximation of the step response with a second order model has

a reasonable fit for values of z ∈ [0.6, 1.4]. Thus, we propose a mid-interval value z1 = 0.9 andreplace it in (A.20) to obtain

Dk+1/Dk =|erf( 1

0.9+2√2σrel

)− 0.5||erf( 1

0.9)− 0.5| , r(σrel).

Then, we can compute the damping factor

r = exp[−πξ

√

1− ξ2]

which yields

ξ2 =log2(r)

π2 + log2(r).

Since (10) states that the period can be approximated as 2/µv, the undamped natural frequency

can now be computed as

ωn =ωd

√

1− ξ2

where

ωd = πµv.

With that, we have the denominator s2 + 2ξωns + ω2n of the transfer function as parametrised by

Corollary 2.

Let us compute the coefficients b0, b1 and b2 of the numerator. We know that the value of D(t)

when the system is in steady state is the expected value of the stochastic expression Ton/(Ton+Toff),

where T ion and Tioff are the times that it takes the i-th AC in the population to go from one end of

the hysteresis band to the other when the device is operating (T ion) or not (Tioff). Since, Ton and

Toff depend on the reference temperature θref , we can write them as a function of θref [24]. Thus

Ton(θref) = CR log

(

PR + θref +H/2− θaPR+ θref −H/2− θa

)

(A.21)

33

and

Toff(θref ) = CR log

(

θa − θref +H/2θa − θref −H/2

)

. (A.22)

Then, replacing these expressions in

Dss(θref) = Ton(θref)/(Ton(θref) + Toff(θref )). (A.23)

we obtain (16).

Let Y (s) = Gp(s)0.5/s in (13) and L−1{Y (s)} = y(t). Then, from the initial value theorem[11], we have

Dss(θref)/2 = y(0+) = lim

s→∞0.5Gp(s) = 0.5b2, (A.24)

therefore b2 = Dss(θref).

On the other hand, from the final value theorem [11], we have

Dss(θref + 0.5)−Dss(θref) = y∞ = lims→0

0.5Gp(s) = 0.5b0/ω2n, (A.25)

therefore b0 = (Dss(θref + 0.5)−Dss(θref ))ω2n/0.5.Combining the initial value theorem and the Laplace transform of a derivative, we have

ẏ(0+) = lims→∞

0.5Gp(s)− 0.5b2 = 0.5(b1 − 2b2ξωn), (A.26)

which gives us b1 = ẏ(0+)/0.5 + 2b2ξωn.

Looking at Figure A.13, we can see that 25% of the ACs are operating at t = 0+. Since we

know that the distribution of x(t) moves to the right at a mean speed µv, we can approximate

ẏ(0+) ≈ y(0++∆t)−y(0+)∆t

= − (0.25−0.25µv∆t)−0.25∆t

= 0.25µv. Thus, b1 = 0.5µv + 2b2ξωn.

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